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1.07 Quantum Numbers
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Do curso por University of Kentucky
Chemistry
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Week 1
We will learn about the components inside the atom and look especially at the electrons. The electron structure determines the properties of the elements. So, by the end of this series of lessons, you will know for each elements where the electrons are located in the atom.
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Dr. Allison SoultLecturer
Chemistry
Dr. Kim WoodrumSr. Lecturer
Chemistry
We are ready for learning objective number seven.
And in this learning objective,
we're going to be examining, the Schrodinger Equation results.
We won't look at the equations themselves, but
we're going to be looking at quantum numbers.
And we're going to see how these quantum lum, numbers tell us
information about where the electron is located, and how much energy it has.
As well as what the electron is doing, how it spins.
So we're going to be examining quantum numbers.
So those complex mathematical equations that define the regions in space,
that define the orbitals, have these four quantum numbers.
There's a principal quantum number n.
The angular momentum quantum number l.
Magnetic quantum number m sub l.
And the spin quantum number m sub s.
Now as we go through and we are examining these four quantum numbers,
I will use addresses as an analogy for what I'm talking about.
We use addresses that define where we live, and
we can get more and more pec, specific with where we live, with the address.
The first address might be what country you live in.
And then maybe you're into a state, or, and then narrow it down to a city, and
then a street address.
Or maybe we could think about a college campus, as a, a broad address.
And then you could talk about what dormitory you live in,
and then you could narrow it down to what room, what is your room number.
And so the first three quantum numbers are giving us the address, getting more and
more specific of where the location of that electron is.
Now, the fourth quantum number, the spin is, it's, spin quantum number is,
what is that electron doing?
The electron is spinning and it can spin one clockwise or counterclockwise.
And so I, you see, statements that the first three tell me where it is,
and the fourth tells me what it's doing.
So we're going to be examining those locations as we go through these four
quantum numbers.
Let's begin with the first principle quantum number, n.
These values of n can go from n equals 1 up to larger and
larger values by whole numbers, integral, integer steps.
One is the smallest.
This is a number we saw with Bohr's model, he called his orbits by these numbers,
n equals 1, n equals 2, n equals 3 and so forth.
And the farther away you got, the higher the n.
Okay, so this is defining the relative distance from the nucleus, and
the energy of the electron.
So the larger the value of n, the further away the electron is.
So if n equals 1 it's close by, nearer the nucleus.
And if n equals 2,
they're a little farther away, n equals 3, they're a little farther away yet.
Now, the farther away the electron is, the higher the energy of that electron.
So as, as n gets larger,
the value of the energy of that electron gets higher as well.
So all electrons with the same n value are in the same principal shell.
So if n equals 1, every electron that has n equals 1 lives in the same shell,
called shell n equals 1, okay.
Principal energy level 1.
Then we can move further away, we have any electron that's located in
principle energy level with an n equals 2, we say they're in the same shell.
Now, for a hydrogen atom, okay, this mathematical equation applies.
If n equals 1, we are located down here and
the electron has this amount of energy down here, 'kay.
So if an electron was sitting here, we would have this much energy.
If n equals 2, it's a little further away, electron is located here and n equals 2.
Or we could have the electron in the n equals 3.
Now, in hydrogen, there’s only one electron.
But if the electrons are located in these different energy levels,
they would have this amount of energy.
And that energy can be, be defined by this equation, what we see here.
The energy, and this is the energy of the electron,
is equal to the Rydberg constant, R sub H times 1 over n squared.
Now, the value of the R sub H is 2.18 times 10 to the minus 18,
and let me get the word out of the way here for you.
10 to the minus 18 joules, and if you
take that value times 1 over n squared, you would have the energy of the electron.
Now we have to have a minus sign here.
Let me def, talk about that a little bit.
It seems weird to have a negative energy.
As we move further and further away, the value is getting less and
less negative, but it is growing in magnitude.
Eventually, you would get up here to where a value of E, the energy, is equal to 0.
Now, at that point, you have just disconnected the electron
from the atom and it is not moving, it has no movement to it.
But as soon as the electron starts moving, then it's going to have
energy associated with its kinetic energy there, and its energy will grow above.
So any time an energy is a positive value for
electrons, that means it's not associated with an atom.
Any time it's a negative value, it's associated with an atom, and
as it gets closer and
closer to the ne, to the nucleus, it has a smaller and smaller energy.
Okay. All right.
Now this Rydberg constant in this equation,
you must understand only applies to a hydrogen atom.
It doesn't apply to anything else, only a hydrogen atom.
It's a much more complex equation and
solution once you put more than one electron in
there because of the interaction the electrons have with each other.
Okay, let's go to the next quantum number.
The next quantum number is the angular momentum quantum number, and
it's abbreviated with an l.
Now, the angular momentum quantum number is related to the principle quantum
number in this way.
An l value can be anywhere from, and in this one, we start with 0 up to
a value of 1 less than your n, up to n minus 1.
This value is going to tell you the shape of the orbital.
And we're going to see the shapes here in a little bit.
But it's going to tell you the general shape of that orbital.
All electrons that have the same n and l value.
Okay, so you have an electron and you tag it.
You say this is where you live.
Okay, n and l, and eventually n sub l.
For every electron that has the same n and
l value, we say that they are in the same subshell.
So if I had one electron that had an n of 2 and an l of 1,
and I had another electron that had an l, n of 2.
Let me write that down, n of 2, and an l equal to 1.
So I have one electron with that established locator, and
I have another electron with that established locator.
We would say those two electrons are in the same subshell.
Now we have names for subshells.
And the subshells correspond to letters.
Those letters match up with l values.
Let's see how that happens.
If an l equals 0, the name of that subshell is s.
If l equals 1 the name is p.
2 is d, 3 is f, and
after that, they go alphabetically, g, h, i, j, k, l, so forth.
You will, as we work through these, not see anything above an f.
But they do exist beyond that.
So these are the names.
So what we would say is, let's say we again have
this electron with an n equal to 2 and an l equal to 1.
Well since the l is equal to 1,
that's called a p subshell, and the p sub, we would say that the electron,
if the n equals 2 and l equals 1, are in the 2p subshell.
All electrons that have n equals 2 and l equals 1 are in the subshell called 2p.
If I had another set of electrons where the n equals 3 and
the l equals 1, then we would say that those are in the 3p subshell.
So all electrons that have the same n and l are in the same subshell.
So that's the name of the subshell.
We're going to see in a little bit when we finally get down to orbitals,
which we're not there yet, that the orbitals are named with these same names.
So each principal level has its own allowable values of l,
because that l can only go up to n minus 1.
You can't have every subshell in every shell, in other words.
So let's look at the possibilities here.
In the shell n equals 4, what are the names of the subshells it has?
Now I want you to stop for just a moment after I do this explanation.
And I want you to think about this.
If n equals 4, the question is, what are the values of l?
Go back and look at how l relates to n.
And once you've done that,
associate names with those, and then choose the right answer.
Pause and resume when you think you know the answer.
Did you pick 4?
Well, if you did, that is correct.
Now, why is that correct?
Let's make sure we're all on board here.
If n equals 4, than we know that l can be 0 up to 1 less than that.
So these are the values of l.
If these are the values of l, this is an s, this is a p,
this is a d, and this is an f, so those, that doesn't look like an f.
Let's try that again.
This is an f.
So those are the four subshells that are in the n equals 4 shell.
So we would call this subshell 4s.
We would call this one 4p.
This one would be 4d, and this one would be 4f.
Those are the four subshells in the fourth shell of any atom.
Okay, let's go to the next quantum number, called m sub l.
It's the magnetic quantum number.
Now the magnetic quantum number is related to the quantum number l.
The magnetic quantum number can go from a negative l up to a positive l.
Now what this piece of information,
it finally gets us down to the actual orbital the electron is in.
Remember, the principle quantum number n,
tells you that general distance away from the nucleus.
The quantum number l is telling you the shape of those orbitals,
and it defines the subshell.
The third one is finally getting down to the actual orbitals themselves.
Now this is going to give you basically the orientation in space.
So the l gave us the general shape and the m sub l is going to say,
okay, how is that in space for that orbital?
We're going to see that here in just a little bit, but let's con,
do our connections.
All electrons with the same n, l, and m sub l are said to be in the same orbital.
So the n is telling you which shell the electron is in.
The l is saying, this is the subshell it's in, and now the m sub l says,
I have narrowed it down to which exact orbital the electron is in.
So we're going to stop here now that we've seen these three quantum numbers n, l,
and m sub l,
and we're going to divine, derive all the possible quantum numbers in a table.
We're just seeing connections and then we will go and actually look at these shapes.
So here's our table.
The smallest n, is an n equals 1.
Now we know that l can go from zero up to n minus 1.
So what are the possibilities for l when n equals 1?
Well, 0 is certainly is one, okay?
So we have 0, can we go any further?
Well, n minus 1 is 0.
So that's as far as we can go.
Now that would be called an s subshell.
So we would be in the 1s subshell if an electron had an n equals 1 and
an l equals 0.
We would be in the 1s subshell.
Let's go to the orbitals.
What are the quantum numbers m sub l?
Now we know that m sub l will go from a negative l
by integral numbers up to a positive l.
Well, if l is 0, I have one choice.
Now when I see that number,
what it tells me is the number of orbitals that are in the 1s subshell.
It doesn't mean that there are zero orbitals, but
however many numbers I see here, is how many orbitals.
Now, I see one number sitting there, so there is one orbital in the 1s subshell.
And we are done.
So let us think about what we have just derived.
What we have said here is that if we are in the first shell, where n equals 1,
there is only one subshell, and that subshell is called the 1s subshell.
And in that subshell, there's only one orbital, and it's called the 1s orbital.
Because they're named the same as their subshell.
So this is one orbital.
It's called a 1s orbital, and that's it.
That's all that's in that first shell.
So it's a very, very, very small city.
Let's move out.
Let's move out a little further away.
So now let's go to n equals 2.
If n equals 2, what can l be?
Well, l goes from 0 up to, and let me do it this way.
Instead of, it would be 0 and it would be 1.
Those are my possibilities.
This would be called a 2s subshell.
This would be called a 2 what?
Well it would be a p because when l equals 1, it's a p.
So those are the two subshells that you have in this shell.
So in the second shell, there's only two subshells, so this is a 2s and
the 2p subshell.
Now, let's move over to the values of n sub l.
For l equals 0, we know that we have only this one.
So there's one orbital again, it's called the 2s orbital.
And then we move to the p value.
The p, when l equals 1, we have from a negative 1 up to a positive 1.
So we have negative 1, 0, and 1.
Now, how many orbitals do you see there?
There are three numbers.
Those three numbers tell me that there are three orbitals in this subshell.
And they are called the 3p orbitals.
I'm sorry.
Three orbitals are called the 2p orbitals.
So I have three of them.
So their names of the orbitals are the same as the names of the shell, and
let's just think about the second shell for a second.
We've come out from the n equals 1, we've moved a little further away.
So we're a little further away out here.
This is the second shell.
In this second shell, there's two subshells.
There's the s and there's the p, okay?
The s subshell only has one orbital.
It's called a 2s orbital.
The p subshell has three orbitals, okay.
They're each called 2p.
They're oriented and spaced differently.
We'll see that in a little bit here, but those are the three p orbitals, 'kay?
By three, I mean there's three of them.
There's three orbitals, each orbital is called a 2p.
Now at any time that you're not catching it, back up and
listen to it again, because this is important that you see these connections.
So, we've gone from the one, first shell to the second shell.
Now we're ready to move out to the third shell.
The third distance away from the nucleus.
Let's do the values of l.
L will go from 0 up to n minus 1.
So we have 0, we have 1 and we would have 2.
'Kay?
So we are going through now,
we've done the first shell, the second shell, the third shell.
In the third shell, you would have a 3s,
you would have a 3p and you would have a 3d subshell.
Okay?
So now let's move to the orbitals.
In the 3s subshell you can have a value of 0, that would be called a 3s orbital.
In the 3p subshell you would go from a negative 1, 0 and 1.
You would have three orbitals there.
Now, let's move out to the d, this is the d subshell.
We would go from a negative 2, up to a positive 2.
So we would negative 2, negative 1, 0, 1, and 2.
How many numbers do you see there?
There are five numbers.
That tells me that we have five d orbitals in a d subshell.
Now this is always true.
If you're in a d subshell,
there will be five orbitals, because we see five numbers.
So this third shell is bigger than the second shell.
It has three subshells, and how many orbitals total does it have?
It hs one, two, three, four, five, six,
seven, eight, nine total orbitals in that third shell.
'Kay, we can move out a little further yet.
Let's go to the n equals 4.
In the n equals 4, we're going to have 0, 1, 2 and 3 for our l values.
That tells me I have a 4s subshell, a 4p subshell,
a 4d subshell, and a 4f subshell.
Those are the subshells that are in the fourth shell.
Now let's look at the orbitals.
When l is 0, we have one number.
When l is 1, we have three numbers.
When l is 2, we have five numbers.
Now, what do we have equal, if l is 3?
Well, we can have an n sub l of a negative 3, negative 2, negative 1, 0, 1, and 2.
We have all of those values.
So, that tells me that there are one, oops.
Gotta get one more.
One, two, three, four, five, six, seven.
There are seven orbitals here.
'Kay, so we have in the f subshell, there are seven orbitals.
So that is the co, all the connections between the quantum numbers.
Now you should be able to keep going further away from that.
But let's just, just think about what these numbers.
Each, each electron in an atom would have these four quantum numbers assigned to it.
It would have an n, an l, and an m sub l assigned to it.
Those four quantum numbers would say, this is where the electron lives.
So lets just pick for quantum numbers that can exist.
Can an electron have an n equals 2, all right.
An l equals 1, that doesn't look like an n equals 2.
Lets try this, can an electron have an n equals 2,
an l equal 1 and an m sub l equal to 0?
Well, let's see.
N equals 2, you can have an l equals 1, and you can, oops, there's 1,
you can have an l equals 1, and you could have an n sub l equal to 0.
So an electron can have those three quantum numbers, and
when you know those three quantum numbers, you know that the electron lives in
an orbital of the p subshell in the second shell.
Could an electron have a value of n equals 2, l equals 2, and n sub l equal to 2?
Well, let's see.
If n equals 2, can you have an l equals 2?
No, because you can only have a 0 and a 1.
So, there is no electron that's going to have that set of quantum numbers.
There is no electron that can live in
a 2d because there's no such thing as a 2d subshell.
Can you think about their connections, and choose which ones is not allowable?
Stop, examine it, and then resume, return.
Okay, so, 2, 2, minus 2, is not allowable.
4, 1, minus 2, is not allowable, so the right answer is both 2 and 4.
Why is that?
What is wrong with them?
Well if you look at this one, if n equals 2, then l can only be 0 and 1.
So I know this is not an allowable l value.
How about this one?
If n equals 4, then l can be 0, 1, 2, or 3.
So this is okay.
We can have a 1 sitting right there.
But if we've narrowed in on to this l value, okay, and we are in a p subshell.
Then the m sub l can only be ze, negative 1, 0 and 1.
It cannot be a negative 2.
So there's the problem with that one.
So you need to know what's allowed and what's not allowed,
based upon the connections between n, l and m sub l, okay?
So if you've had an electron, and you knew this electron was in the 4p subshell,
which one of those quantum numbers could be the address of that electron?
Stop and look at those, and then resume.
So we see that the right answer is, both 2 and 3, and so
if you picked that, you were correct.
Let's examine why that would be.
This number here tells you n.
This number tells you l, 'kay?
So we know that n has to be 4.
So all of those have an n of 4.
We can't narrow any down there.
But if l, if it's a p subshell, we know that when l is 0, that's an s.
When l is 1, that's a p.
So l has to be 1.
So that's this guy and this guy.
The question is, are those allowable n sub l's for that l?
Well, if l equals 1, we know that n sub l could be negative 1, 0, or 1.
So either one of these would be fine.
There would be one other one that would have been okay to write, and
that would be 4, 1, and 1.
That would be fine as well.
The electron could have that address.
So, any one of those values would be acceptable for
an electron in a 4p subshell.
'Kay, so, this is the end of Learning Objective number seven.
We've examined the quantum numbers and
seen the allowable quantum numbers that tell you the location of your electron.
So an electron has these three taggers of n, l, and m sub l.
Now we haven't talked about m sub s yet.
We will here in a little bit.
But these three numbers tell you the location, n is the first distance away.
If, if n is 1 it's closer, 2 it's a little further, 3 it's a little further yet.
M sub l tells you the subshell in each shell, 'kay?
So it's going to be telling you basically the shape.
We haven't seen the shapes yet, we will here in a little while.
And m sub l tells you how many orbitals are in that subshell and
they will define the orientation in space, which we will also see in a little bit.
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